$L^p$-BOUNDEDNESS OF THE BEREZIN TRANSFORM ON GENERALISED HARTOGS TRIANGLES

Abstract We study the $L^p$ -boundedness of the Berezin transform on the generalised Hartogs triangles which are defined by $$ \begin{align*}H_k:=\{(z, w)\in\mathbb C^n\times\mathbb C: |z_1|^2+\cdots+|z_n|^2<|w|^{2k}<1\},\end{align*} $$ where $z=(z_1, \ldots , z_n)$ and $k\in \mathbb N$ . We prove that the Berezin transform is bounded on $L^p(H_k)$ if and only if $p>nk+1$ .